Graph spectral analysis/Graph spectral clustering and its application to metabolic networks

Graph spectral analysis/

Graph spectral clustering

PROTEIN STRUCTURE: INSIGHTS FROM GRAPH THEORY

SARASWATHI VISHVESHWARA, K. V. BRINDA and N. KANNANyMolecular Biophysics Unit, Indian Institute of Science

Bangalore 560012, India

Laplacian matrix L=D-A

Adjacency Matrix Degree Matrix

Eigenvalues of a matrix A are the roots of the following equation

|A-λI|=0, where I is an identity matrix

Let λ is an eigenvalue of A and x is a vector such that

then x is an eigenvector of A corresponding to λ .

-----(1)N×N N×1 N×1

Eigenvalues and eigenvectors

Node 1 has 3 edges, nodes 2, 3 and 4 have 2 edges each and node 5 has only one edge. The magnitude of the vector components of the largest eigenvalue of the Adjacency matrix reflects this observation.

Node 1 has 3 edges, nodes 2, 3 and 4 have 2 edges each and node 5 has only one edge. The magnitude of the vector components of the largest eigenvalue of the Laplacian matrix reflects this observation.

The largest eigenvalue (lev) depends upon the highest degree in the graph. For any k regular graph G (a graph with k degree on all the vertices), the eigenvalue with the largest absolute value is k. A corollary to this theorem is that the lev of a clique of n verticesis n − 1. In a general connected graph, the lev is always ≤ to the largest degree in the graph. In a graph with n vertices, the absolute value of lev decreasesas the degree of vertices decreases. The lev of a clique with 11 vertices is 10 and that of a linearchain with 11 vertices is 1.932

a linear chain with 11 vertices

In graphs 5(a)-5(e), the highest degree is 6. In graphs 5(f)-5(i), the highest degree is 5, 4, 3 and 2 respectively.

It can be noticed that the lev is generally higher if the graph contains vertices of high degree. The lev decreases gradually from the graph with highest degree 6 to the one with highest degree 2. In case of graphs 5(a){5(e), where there is one common vertex with degree 6 (highest degree) and the degrees on the other vertices are different (less than 6 in all cases), the lev also depends on the degree of the vertices adjoining the highest degree vertex.

We combine graph 4(a) and graph 4(b) and construct a Laplacian matrix with edge weights (1/dij ), where dij is the distance between vertices i and j. The distances between the vertices of graph 4(a) and graph 4(b) are considered to be very large (say 100) and thus the matrix elements corresponding to a vertex from graph 4(a) and the other from graph 4(b) is considered to have a very small value of 0.01. The Laplacian matrix of 8 vertices thus considered is diagonalized and their eigenvalues and corresponding vector components are given in Table 3.

The vector components corresponding to the second smallest eigenvalue contains the desired information about clustering, where the cluster forming residues have identical values. In Fig. 4, nodes 1-5 form a cluster (cluster 1) and 6-8 form another cluster (cluster 2).

Metabolome Based Reaction Graphs of M. tuberculosisand M. leprae: A Comparative Network AnalysisKetki D. Verkhedkar1, Karthik Raman2, Nagasuma R. Chandra2, Saraswathi Vishveshwara1*1 Molecular Biophysics Unit, Indian Institute of Science, Bangalore, India, 2 Bioinformatics Centre, Supercomputer Education and Research Centre, Indian Institute of Science, Bangalore, IndiaPLoS ONE | www.plosone.org September 2007 | Issue 9 | e881

Construction of network

R1 R2

R3 R4

Analysis of network parameters

Analyses of sub-clusters in the giant componentTo detect sub-clusters of reactions in the giant component graph spectral analysis was performed.To obtain the eigenvalue spectra of the graph, the adjacency matrix of the graph is converted to a Laplacian matrix (L), by the equation:L=D-Awhere D, the degree matrix of the graph, is a diagonal matrix in which the ith element on the diagonal is equal to the number of connections that the ith node makes in the graph.

It is observed that reactions belonging to fatty acid biosynthesis and the FAS-II cycle of the mycolic acid pathway in M. tuberculosis form distinct, tightly connected sub-clusters.

Identification of hubs in the reaction networksIn biological networks, the hubs are thought to be functionally important and phylogenetically oldest.

The largest vector component of the highest eigenvalue of the Laplacian matrix of the graph corresponds to the node with high degree as well as low eccentricity. Two parameters, degree and eccentricity, are involved in the identification of graph spectral (GS) hubs.Alternatively, hubs can be ranked based on their connectivity alone (degree hubs).

It was observed that the top 50 degree hubs in the reaction networks of the three organisms comprised reactions involving the metabolite L-glutamate as well as reactions involving pyruvate. However, the top 50 GS hubs of M. tuberculosis and M. leprae exclusively comprised reactions involving L-glutamate while the top GS hubs in E. coli only consisted of reactions involving pyruvate.

The difference in the degree and GS hubs suggests that the most highly connected reactions are not necessarily the most central reactions in the metabolome of the organism